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| Mirrors > Home > MPE Home > Th. List > domentr | Structured version Visualization version GIF version | ||
| Description: Transitivity of dominance and equinumerosity. (Contributed by NM, 7-Jun-1998.) |
| Ref | Expression |
|---|---|
| domentr | ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | endom 9019 | . 2 ⊢ (𝐵 ≈ 𝐶 → 𝐵 ≼ 𝐶) | |
| 2 | domtr 9047 | . 2 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶) | |
| 3 | 1, 2 | sylan2 593 | 1 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐵 ≈ 𝐶) → 𝐴 ≼ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 class class class wbr 5143 ≈ cen 8982 ≼ cdom 8983 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-f1o 6568 df-en 8986 df-dom 8987 |
| This theorem is referenced by: domdifsn 9094 xpdom1g 9109 domunsncan 9112 sdomdomtr 9150 domen2 9160 mapdom2 9188 phpOLD 9259 unxpdom2 9290 sucxpdom 9291 xpfir 9300 fodomfiOLD 9370 cardsdomelir 10013 infxpenlem 10053 xpct 10056 infpwfien 10102 inffien 10103 mappwen 10152 iunfictbso 10154 djuxpdom 10226 cdainflem 10228 djuinf 10229 djulepw 10233 ficardun2 10242 unctb 10244 infdjuabs 10245 infunabs 10246 infdju 10247 infdif 10248 infxpdom 10250 pwdjudom 10255 infmap2 10257 fictb 10284 cfslb 10306 fin1a2lem11 10450 fnct 10577 unirnfdomd 10607 iunctb 10614 alephreg 10622 cfpwsdom 10624 gchdomtri 10669 canthp1lem1 10692 pwfseqlem5 10703 pwxpndom 10706 gchdjuidm 10708 gchxpidm 10709 gchpwdom 10710 gchhar 10719 inttsk 10814 inar1 10815 tskcard 10821 znnen 16248 qnnen 16249 rpnnen 16263 rexpen 16264 aleph1irr 16282 cygctb 19910 1stcfb 23453 2ndcredom 23458 2ndcctbss 23463 hauspwdom 23509 tx2ndc 23659 met1stc 24534 met2ndci 24535 re2ndc 24822 opnreen 24853 ovolctb2 25527 ovolfi 25529 uniiccdif 25613 dyadmbl 25635 opnmblALT 25638 vitali 25648 mbfimaopnlem 25690 mbfsup 25699 aannenlem3 26372 dmvlsiga 34130 sigapildsys 34163 omssubadd 34302 carsgclctunlem3 34322 finminlem 36319 phpreu 37611 lindsdom 37621 mblfinlem1 37664 pellexlem4 42843 pellexlem5 42844 pr2dom 43540 tr3dom 43541 nnfoctb 45053 ioonct 45550 subsaliuncl 46373 caragenunicl 46539 aacllem 49320 |
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